#72321135

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#151085
commutator	C₃₂	C₂³×C₈
Frattini, cosocle	#1397	C₂⁴
socle	C₂	C₂×#151085
prime radical	#513315	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(1,2)(3,16)(4,23)(5,31)(6,36)(7,10)(8,11)(9,18)(12,28)(13,26)(14,34)(15,38)(17,61)(19,60)(20,70)(21,58)(22,59)(24,54)(25,77)(27,74)(29,94)(30,95)(32,51)(33,98)(35,102)(37,57)(39,47)(40,62)(41,67)(42,68)(43,82)(44,80)(45,92)(46,89)(48,90)(49,86)(50,101)(52,116)(53,84)(55,83)(56,66)(63,119)(64,120)(65,104)(69,117)(71,111)(72,114)(73,124)(75,87)(76,93)(78,118)(79,108)(81,113)(85,126)(88,97)(91,128)(96,105)(99,109)(100,115)(103,121)(106,125)(107,123)(110,122)(112,127);(1,3)(2,9)(4,24)(5,32)(6,37)(7,19)(8,21)(10,42)(11,41)(12,51)(13,54)(14,40)(15,56)(16,23)(17,34)(18,28)(20,26)(22,38)(25,79)(27,81)(29,78)(30,96)(31,43)(33,71)(35,99)(36,44)(39,63)(45,109)(46,105)(47,104)(48,111)(49,113)(50,103)(52,117)(53,115)(55,108)(57,94)(58,74)(59,77)(60,95)(61,76)(62,116)(64,101)(65,122)(66,84)(67,90)(68,92)(69,89)(70,75)(72,86)(73,83)(80,87)(82,93)(85,125)(88,100)(91,123)(97,118)(98,106)(102,107)(110,114)(112,124)(119,128)(120,126)(121,127);(2,10,11,15)(3,17,19,63,21,64,22,37)(4,25,27,30)(5,33)(9,40,42,104,41,103,56,44)(12,48)(13,55,49,46)(14,47,50,36)(16,57,59,120,58,119,60,61)(18,65,66,62,67,80,68,121)(20,71,69,100,72,32,73,99)(24,78,79,125,81,123,96,82)(26,86)(28,92,90,84)(29,85,91,93)(31,97,98,102)(35,88)(43,108,107,54,106,105,118,113)(45,53)(51,114,115,117,111,70,109,124)(52,112)(75,110)(83,89)(87,116,122,127);(2,11)(9,41)(10,15)(12,48)(13,49)(14,50)(16,58)(23,74)(31,98)(36,47)(40,103)(42,56)(43,106)(44,104)(45,53)(46,55)(51,111)(52,112)(54,113)(57,119)(59,60)(61,120)(70,114)(75,110)(76,126)(77,95)(94,128)(97,102)(105,108)(107,118)(109,115)(117,124)

Polycyclic presentation:

2^11|c^2=f,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=be,c^a=cj,c^b=cij,d^a=dk,e^a=eg,e^b=eg,e^c=ejk,f^a=fk,f^b=fj,f^e=fk,g^a=gi,g^b=gi,g^c=gk,h^b=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321290, #72321544, #72321722, …
maximal normal subgroup of	#72321290, #72321544, #72321722, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);